Problem 45 - Algebra and Trigonometry

Question

\(43-46\). Write the given expression as an algebraic expression in \(x\). $$ \sin \left(\frac{1}{2} \cos ^{-1} x\right) $$

Step-by-Step Solution

Answer

\( \sqrt{\frac{1-x}{2}} \)

Step 1: Understand the problem

We need to rewrite \( \sin \left(\frac{1}{2} \cos^{-1} x\right) \) as an algebraic expression. This involves converting it to a form in terms of \( x \) only, without trigonometric functions.

Step 2: Use trigonometric identity

Recall that \( \sin \left(\frac{1}{2} \theta\right) = \sqrt{\frac{1-\cos \theta}{2}} \). We will apply this identity to the expression \( \sin \left(\frac{1}{2} \cos^{-1} x\right) \).

Step 3: Substitute and simplify

Substitute \( \theta = \cos^{-1} x \) into the identity from Step 2. This gives us \( \sin \left(\frac{1}{2} \cos^{-1} x\right) = \sqrt{\frac{1-\cos(\cos^{-1} x)}{2}} \).

Step 4: Simplify further

Note that \( \cos(\cos^{-1} x) = x \). Thus, the expression becomes \( \sqrt{\frac{1-x}{2}} \).

Step 5: Final expression

The algebraic expression for \( \sin \left(\frac{1}{2} \cos^{-1} x\right) \) is \( \sqrt{\frac{1-x}{2}} \).

Key Concepts

Trigonometric Identity

Trigonometric identities are equations that are true for all values of the occurring variables where both sides of the equation are defined. These identities play a crucial role not only in mathematics but also in physics, engineering, and various fields of science. In this exercise, we primarily use the identity for the sine of a half-angle:

• \( \sin \left( \frac{1}{2} \theta \right) = \sqrt{\frac{1-\cos \theta}{2}} \)

This identity helps us transform trigonometric expressions into algebraic ones, which are often easier to work with, especially in solving equations or simplifying expressions. It allows us to convert the given trigonometric function involving arccosine into a simpler form dependent on \( x \). Such transformations are integral in calculus and analysis, especially during integration and differentiation of trigonometric functions.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the basic trigonometric functions. They are used to find the angle that corresponds to a given trigonometric value. The function \( \cos^{-1}(x) \), also known as arccosine, returns the angle whose cosine is \( x \).

• The domain of \( \cos^{-1}(x) \) is \([-1, 1]\), and its range is \([0, \pi]\).

When working with inverse functions, we often use them to change a problem from the trigonometric form to an angle form, permitting easier manipulation. In our exercise, substituting \( \theta = \cos^{-1} x \) alters the context from trigonometric terms to angular terms, aiding simplification through known trigonometric identities.
For instance, replacing \( \theta \) within \( \sin \left(\frac{1}{2} \theta \right) \) enables the application of the half-angle identity, aiding us to transition from inverse trigonometric function efficiently to an algebraic expression.

Simplifying Expressions

Simplifying expressions involves rewriting them in a more straightforward or standardized form. It is a critical skill in mathematics, allowing us to understand and work with equations more effectively.

• In the context of our given expression \( \sin \left( \frac{1}{2} \cos^{-1} x \right) \), simplification began by using identities and substitutions that reduce complexity.

The first key simplification step involved substituting the inverse function with its equivalent, \( \theta = \cos^{-1} x \), which used the understanding that \( \cos(\cos^{-1} x) = x \).
After applying the half-angle identity, the expression further simplified to \( \sqrt{\frac{1-x}{2}} \).
Each step systematically reduced complexity, ultimately turning a trigonometric expression into a neat algebraic form.
This process illustrates how understanding identities and inverse functions contributes significantly to the simplification of mathematical expressions, making them easier to interpret and solve.